1.02a Indices: laws of indices for rational exponents

3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).

2 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 2 } { x ^ { 3 } }\) passes through \(( - 1,3 )\). Find the equation of the curve.

2 Solve the equation \(x ^ { 3.9 } = 11 x ^ { 3.2 }\), where \(x \neq 0\).

2

1. Express \(4 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\).
2. Hence find the values of \(x\) that satisfy the equation $$3 \left( 4 ^ { x } \right) - 10 \left( 2 ^ { x } \right) + 3 = 0 ,$$ giving your answers correct to 2 decimal places.

1. Given

$$\frac { 3 ^ { x } } { 3 ^ { 4 y } } = 27 \sqrt { 3 }$$ find \(y\) as a simplified function of \(x\).

2. Given \(y = 3 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.

1. \(3 ^ { 3 x }\)
2. \(\frac { 1 } { 3 ^ { x - 2 } }\)
3. \(\frac { 81 } { 9 ^ { 2 - 3 x } }\)

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 2 ^ { x }\), show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
2. Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$

2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = a x ^ { 3 } + ( 6 a + 8 ) x ^ { 2 } - a ^ { 2 } x$$ where \(a\) is a positive constant. Given \(\mathrm { f } ( - 1 ) = 32\)

1. 1. show that the only possible value for \(a\) is 3
   2. Using \(a = 3\) solve the equation $$\mathrm { f } ( x ) = 0$$
2. Hence find all real solutions of
   1. \(3 y + 26 y ^ { \frac { 2 } { 3 } } - 9 y ^ { \frac { 1 } { 3 } } = 0\)
   2. \(3 \left( 9 ^ { 3 z } \right) + 26 \left( 9 ^ { 2 z } \right) - 9 \left( 9 ^ { z } \right) = 0\)

1. Given that

$$\left( 3 p q ^ { 2 } \right) ^ { 4 } \times 2 p \sqrt { q ^ { 8 } } \equiv a p ^ { b } q ^ { c }$$ find the values of the constants \(a , b\) and \(c\).

1. The share price of a company is monitored.

Exactly 3 years after monitoring began, the share price was \(\pounds 1.05\) Exactly 5 years after monitoring began, the share price was \(\pounds 1.65\) The share price, \(\pounds V\), of the company is modelled by the equation $$V = p t + q$$ where \(t\) is the number of years after monitoring began and \(p\) and \(q\) are constants.

1. Find the value of \(p\) and the value of \(q\). Exactly \(T\) years after monitoring began, the share price was \(\pounds 2.50\)
2. Find the value of \(T\), according to the model, giving your answer to one decimal place.

1. Given that

$$y = 5 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } - 7 x \quad x > 0$$ find, in simplest form,

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)

Simplify the following expressions fully.

1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)

2. (i) Given that \(\frac { 49 } { \sqrt { 7 } } = 7 ^ { a }\), find the value of \(a\).
(ii) Show that \(\frac { 10 } { \sqrt { 18 } - 4 } = 15 \sqrt { 2 } + 20\) You must show all stages of your working.

12. $$f ( x ) = \frac { ( 4 + 3 \sqrt { } x ) ^ { 2 } } { x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = A x ^ { - 1 } + B x ^ { k } + C\), where \(A , B , C\) and \(k\) are constants to be determined.
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\).
3. Find an equation of the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 4\) 2. LIIIII

3. Simplify fully

1. \(\left( 3 x ^ { \frac { 1 } { 2 } } \right) ^ { 4 }\)
2. \(\frac { 2 y ^ { 7 } \times ( 4 y ) ^ { - 2 } } { 3 y }\)

2. Given \(y = 2 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.

1. \(2 ^ { 2 x }\)
2. \(2 ^ { x + 3 }\)
3. \(\frac { 1 } { 4 ^ { 2 x - 3 } }\)

3. Given that $$y = \frac { 1 } { 27 } x ^ { 3 }$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.

1. \(y ^ { \frac { 1 } { 3 } }\)
2. \(3 y ^ { - 1 }\)
3. \(\sqrt { ( 27 y ) }\)

3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form.

1. Solve the equation $$4 ^ { 2 x + 1 } = 8 ^ { 4 x }$$
2. (a) Express $$3 \sqrt { 18 } - \sqrt { 32 }$$ in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
   (b) Hence, or otherwise, solve $$3 \sqrt { 18 } - \sqrt { 32 } = \sqrt { n }$$

6. (a) Show that \(\frac { x ^ { 2 } - 4 } { 2 \sqrt { } x }\) can be written in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants to be determined.
(b) Hence find $$\int \frac { x ^ { 2 } - 4 } { 2 \sqrt { x } } \mathrm {~d} x , \quad x > 0$$ giving your answer in its simplest form.

4. Given that $$y = \frac { 64 x ^ { 6 } } { 25 } , x > 0$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.

1. \(y ^ { - \frac { 1 } { 2 } }\)
2. \(( 25 y ) ^ { \frac { 2 } { 3 } }\)

1. (i) Given that \(125 \sqrt { 5 } = 5 ^ { a }\), find the value of \(a\).
   (ii) Show that \(\frac { 16 } { 4 - \sqrt { 8 } } = 8 + 4 \sqrt { 2 }\)

You must show all stages of your working.